Question

for the function ( f(x)=ln(1 + 4x) ), find the taylor polynomials of orders 0, 1, 2, and 3 generated by ( f ) at ( a = 0 ). ( p_0(x)=square )

Explanation:

Step1: Recall the formula for Taylor polynomial of order \(n\)

The Taylor polynomial of order \(n\) for a function \(f(x)\) about \(a\) is \(P_{n}(x)=\sum_{k = 0}^{n}\frac{f^{(k)}(a)}{k!}(x - a)^{k}\). When \(a = 0\), it is \(P_{n}(x)=\sum_{k=0}^{n}\frac{f^{(k)}(0)}{k!}x^{k}\), and for \(n = 0\), \(P_{0}(x)=f(0)\).

Step2: Evaluate \(f(0)\)

Given \(f(x)=\ln(1 + 4x)\), substitute \(x = 0\) into \(f(x)\). Then \(f(0)=\ln(1+4\times0)=\ln(1)=0\).

Answer:

\(0\)